Textbook Question
Finding Extreme Values
In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.
y = 𝓍³ ― 2𝓍 + 4
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.7.33
Finding Indefinite Integrals
In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫(t√t + √t) / t² dt
Verified step by step guidance1
First, simplify the integrand by expressing all terms with exponents. Recall that \( \sqrt{t} = t^{1/2} \) and rewrite the expression inside the integral accordingly.
Rewrite the integrand \( \frac{t\sqrt{t} + \sqrt{t}}{t^2} \) as \( \frac{t \cdot t^{1/2} + t^{1/2}}{t^2} = \frac{t^{3/2} + t^{1/2}}{t^2} \).
Split the fraction into two separate terms: \( \frac{t^{3/2}}{t^2} + \frac{t^{1/2}}{t^2} \). Then simplify each term by subtracting exponents in the denominator from those in the numerator.
After simplification, express the integrand as a sum of powers of \( t \): \( t^{3/2 - 2} + t^{1/2 - 2} = t^{-1/2} + t^{-3/2} \).
Now, integrate each term separately using the power rule for integration: \( \int t^n dt = \frac{t^{n+1}}{n+1} + C \), making sure to add the constant of integration at the end.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Indefinite Integral and Antiderivative
An indefinite integral represents the most general antiderivative of a function, including a constant of integration. It reverses differentiation, finding a function whose derivative matches the integrand. Understanding this helps in solving integrals without specified limits.
Recommended video:
05:04
Introduction to Indefinite Integrals
Algebraic Simplification of the Integrand
Before integrating, simplify the integrand by combining like terms and rewriting expressions using exponent rules. For example, rewrite roots as fractional exponents and simplify fractions to make integration straightforward and reduce errors.
Recommended video:
05:22Completing the Square to Rewrite the Integrand
Power Rule for Integration
The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C for any real number n ≠ -1. This rule is fundamental for integrating polynomial and power functions, allowing direct computation once the integrand is expressed in terms of powers.
Recommended video:
04:04Power Rule for Indefinite Integrals
Related Practice
Textbook Question
Finding Indefinite Integrals
Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫ sec² s/10 ds
Textbook Question
Applications
Liftoff from Earth A rocket lifts off the surface of Earth with a constant acceleration of 20 m/sec². How fast will the rocket be going 1 min later?
Textbook Question
Identifying Extrema
In Exercises 19–40:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local extreme values, if any, saying where they occur.
f(x) = (x² − 3) / (x − 2), x ≠ 2
Textbook Question
Identifying Extrema
In Exercises 15–18:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local and absolute extreme values, if any, saying where they occur.
Textbook Question
54. Fermat’s principle in optics Light from a source A is reflected by a plane mirror to a receiver at point B, as shown in the accompanying figure. Show that for the light to obey Fermat’s principle, the angle of incidence must equal the angle of reflection, both measured from the line normal to the reflecting surface. (This result can also be derived without calculus. There is a purely geometric argument, which you may prefer.)
1
views